Question: Simplify and expand the following expression: $ \dfrac{4}{2a - 4}+ \dfrac{4}{a - 10}+ \dfrac{4a}{a^2 - 12a + 20} $
Solution: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the first term: $ \dfrac{4}{2a - 4} = \dfrac{4}{2(a - 2)}$ We can factor the quadratic in the third term: $ \dfrac{4a}{a^2 - 12a + 20} = \dfrac{4a}{(a - 2)(a - 10)}$ Now we have: $ \dfrac{4}{2(a - 2)}+ \dfrac{4}{a - 10}+ \dfrac{4a}{(a - 2)(a - 10)} $ The least common multiple of the denominators is: $ 2(a - 2)(a - 10)$ In order to get the first term over $2(a - 2)(a - 10)$ , multiply by $\dfrac{a - 10}{a - 10}$ $ \dfrac{4}{2(a - 2)} \times \dfrac{a - 10}{a - 10} = \dfrac{4(a - 10)}{2(a - 2)(a - 10)} $ In order to get the second term over $2(a - 2)(a - 10)$ , multiply by $\dfrac{2(a - 2)}{2(a - 2)}$ $ \dfrac{4}{a - 10} \times \dfrac{2(a - 2)}{2(a - 2)} = \dfrac{8(a - 2)}{2(a - 2)(a - 10)} $ In order to get the third term over $2(a - 2)(a - 10)$ , multiply by $\dfrac{2}{2}$ $ \dfrac{4a}{(a - 2)(a - 10)} \times \dfrac{2}{2} = \dfrac{8a}{2(a - 2)(a - 10)} $ Now we have: $ \dfrac{4(a - 10)}{2(a - 2)(a - 10)} + \dfrac{8(a - 2)}{2(a - 2)(a - 10)} + \dfrac{8a}{2(a - 2)(a - 10)} $ $ = \dfrac{ 4(a - 10) + 8(a - 2) + 8a} {2(a - 2)(a - 10)} $ Expand: $ = \dfrac{4a - 40 + 8a - 16 + 8a}{2a^2 - 24a + 40} $ $ = \dfrac{20a - 56}{2a^2 - 24a + 40}$ Simplify: $ = \dfrac{10a - 28}{a^2 - 12a + 20}$